behavior of rigid and deformable particles in deterministic lateral … · 2018. 6. 19. · preface...

Behavior of rigid and deformable particles in deterministic lateral displacementdevices with different post shapesZunmin Zhang, Ewan Henry, Gerhard Gompper, and Dmitry A. Fedosov Citation: The Journal of Chemical Physics 143, 243145 (2015); doi: 10.1063/1.4937171 View online: http://dx.doi.org/10.1063/1.4937171 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/143/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Deformability-based red blood cell separation in deterministic lateral displacement devices—A simulationstudy Biomicrofluidics 8, 054114 (2014); 10.1063/1.4897913 Rapid isolation of cancer cells using microfluidic deterministic lateral displacement structure Biomicrofluidics 7, 011801 (2013); 10.1063/1.4774308 Shape controllable microgel particles prepared by microfluidic combining external ionic crosslinking Biomicrofluidics 6, 026502 (2012); 10.1063/1.4720396 Preface to Special Topic: Microsystems for manipulation and analysis of living cells Biomicrofluidics 5, 031901 (2011); 10.1063/1.3641860 Observation of nonspherical particle behaviors for continuous shape-based separation using hydrodynamicfiltration Biomicrofluidics 5, 024103 (2011); 10.1063/1.3580757

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THE JOURNAL OF CHEMICAL PHYSICS 143, 243145 (2015)

Behavior of rigid and deformable particles in deterministic lateraldisplacement devices with different post shapes

Zunmin Zhang, Ewan Henry, Gerhard Gompper, and Dmitry A. Fedosova)Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation,Forschungszentrum Jülich, 52425 Jülich, Germany

(Received 20 September 2015; accepted 23 November 2015; published online 14 December 2015)

Deterministic lateral displacement (DLD) devices have great potential for the separation and sortingof various suspended particles based on their size, shape, deformability, and other intrinsic properties.Currently, the basic idea for the separation mechanism is that the structure and geometry of DLDsuniquely determine the flow field, which in turn defines a critical particle size and the particle lateraldisplacement within a device. We employ numerical simulations using coarse-grained mesoscopicmethods and two-dimensional models to elucidate the dynamics of both rigid spherical particles anddeformable red blood cells (RBCs) in different DLD geometries. Several shapes of pillars, includingcircular, diamond, square, and triangular structures, and a few particle sizes are considered. Thesimulation results show that a critical particle size can be well defined for rigid spherical particlesand depends on the details of the DLD structure and the corresponding flow field within the device.However, non-isotropic and deformable particles such as RBCs exhibit much more complex dynamicswithin a DLD device, which cannot properly be described by a single parameter such as the criticalsize. The dynamics and deformation of soft particles within a DLD device become also important,indicating that not only size sorting, but additional sorting targets (e.g., shape, deformability, internalviscosity) are possible. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4937171]

I. INTRODUCTION

The deterministic lateral displacement (DLD) approach1

for particle and cell sorting has attracted considerable researchand technological interest for developing novel microfluidicdevices for biological and clinical applications. Starting fromits invention by Huang et al.2 as a size-dependent (label-free) particle-separation method, this technique has beencontinuously used and developed further. DLD devices arenow able to separate multiple particles with an extremelyhigh resolution of about 20 nm when applied to the sortingof polystyrene beads.2–4 Other applications also include theseparation of parasites from human blood,5 fractionation ofhuman blood cells,6,7 and sorting of droplets in microfluidics.8

This demonstrates a great potential of DLDs for the separationand detection of various rigid and deformable particles in asuspension based on particle properties such as size, shape,deformability, and potentially other intrinsic characteristics.However, a rational design of such devices requires a detailedunderstanding of the flow behavior of different particles incomplex geometrical DLD structures.

In general, a DLD device consists of a row-shifted arrayof micron-sized pillars (or posts) residing in the flow channel.As shown in Fig. 1, the posts are arranged in different rowsand each row is shifted laterally with respect to the previousone. The geometry and arrangement of posts (e.g., post shape,post gap G, row shift ∆λ) determine a critical particle sizeDc for the DLD device. According to the theory,2 particles

a)Author to whom correspondence should be addressed. Electronic mail:[email protected]

smaller than Dc tend to move in the direction of flow known as“zigzag mode,” whereas particles larger than Dc are laterallydisplaced and move along the post array gradient known as“displacement mode.” In this way, different particles can beseparated by collecting them at different outlets of a DLDdevice and thus, the knowledge about critical size allows thedesign of devices for sorting specific sizes of rigid sphericalparticles. Recently, a third motion named “mixed mode”has been observed in experiments and simulations,9,10 whichshows an irregular alternation of zigzag and displacementmodes. The erratic nature of the mixed mode may smearthe otherwise deterministic nature of a DLD device andconsequently complicate device design and application.

The deterministic separation mechanism described abovefunctions well for rigid spherical particles whose sizes remainconstant under various experimental conditions (e.g., flowrate). In practice, it is desirable to separate more exoticparticles with different rigidities and shapes; for instance, redblood cells (RBCs) are soft and non-spherical. Particles withvarious rigidities will deform to a different extent under theshear forces experienced in flow. Examples include vesicles instructured microchannels,11 RBCs in cylindrical vessels,12–15

and synthetic microcapsules in shear flow.16–18 Thus, the effec-tive size of soft particles will vary depending on their locationin the complex flow field within the post array. Furthermore,particles with non-isotropic shapes have varying effectivesize depending on their orientation relative to the post array.For example, Beech et al.19 have presented an orientation-and deformation-based separation of RBCs achieved by thereduction of the channel depth and the change of flow ratewithin the device, respectively. These two uncertainties in

0021-9606/2015/143(24)/243145/11/$30.00 143, 243145-1 © 2015 AIP Publishing LLC


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243145-2 Zhang et al. J. Chem. Phys. 143, 243145 (2015)

FIG. 1. Schematic illustration of the geometry and structures of simulatedDLD systems. (a) Typical post shapes (circle, square, diamond, and triangle)with W =H = 15 µm. (b) DLD array with circular posts. The geometryis defined by the post center-to-center distance λ = 25 µm, the post gapG = 10 µm, and the row shift ∆λ. The red arrows indicate the direction offlow. (c) 2D models of RBCs and rigid circular particles.

effective size do not necessarily destroy the determinism of aDLD device, but they certainly make it harder to predict whichmode of motion a particle will adopt in transit through a DLDdevice. In addition, there is no clear evidence that conven-tional circular post arrays are the best option, which makesthe geometry of posts an important additional parameter toexplore. Zeming et al.20 have created a DLD device withI-shaped posts in order to induce RBC rotation, which conse-quently increases the hydrodynamic size of RBCs and alterstheir behavior within the DLD device. Al-Fandi et al.21 haveproposed airfoil-like and diamond-like post shapes and foundthat the airfoil post array appears to be better for the reductionof clogging and deformation of soft particles. Loutherbacket al.3,4 have demonstrated that the use of a triangular postarray can significantly enhance the performance of DLDdevices by reducing their clogging, lowering hydrostatic pres-sure requirements, and increasing the range of displacement-type dynamics. In view of these studies, post shape plays animportant role in controlling the separation effectiveness ofvarious particles by modifying the flow profiles within DLDarrays and introducing novel particle trajectories.

Attempting to experimentally realize the transit modes forthe myriad of combinations of particle shapes and properties,device geometries, and post shapes would be very expensiveand extremely time-consuming. Simulations provide avaluable alternative, allowing systematic exploration ofparticle transit behavior through novel post-array geometries.Several numerical simulations22–24 have investigated thebehavior of elastic capsules and RBCs in DLD deviceswith circular post arrays indicating that an effective diameterof a deformable particle may change with the flow rate

or equivalently that the deformation of such particlesdepends on the stresses and profile of microfluidic flow.In this study, we employ coarse-grained methods and two-dimensional (2D) models to simulate the motion of rigidspherical particles and deformable RBCs in DLD deviceswith different post geometries. Since such DLD geometriesare at the microscale, simulation techniques on atomic ormolecular levels become unfeasible. Therefore, we employa coarse-grained hydrodynamics method at the mesoscalecalled dissipative particle dynamics,25,26 allowing us to accessmuch longer length and times scales through a coarse fluidrepresentation. The models for suspended particles are also ona much coarser level than a representation on the molecularscale. Coarse-grained simulation approaches are ideally suitedfor such an investigation, because they not only allow to reachthe required length- and time scales but also make it possibleto abstract from atomistic and molecular detail, which isirrelevant for the design of devices. Another simplification isthe use of 2D simulations, which needs to be exercised withcaution, since fluid flow and the motion and deformation ofsoft particles in microfluidics is inherently three-dimensional(3D). There are several simulation studies, such as motionof RBCs in microchannels15,27 and the margination of whiteblood cells in blood flow,28–30 which have shown that 2Dand 3D simulations lead to qualitatively consistent results.This supports the idea that 2D models should also capture theessential physics of particle separation in DLDs and providevaluable insights. The main advantage of 2D simulationsin comparison to 3D models is a comparatively lowcomputational cost, which allows for systematic explorationof many relevant parameters in DLDs.

As an initial proof of the method, we compare simulationresults for hard spherical particles in a circular-post array withexperimental results. We demonstrate by this comparison thatfor this case, 2D simulations provide near quantitative resultsdue to the symmetry of spherical particles. This is not anobvious result, since the flow field near a spherical particle in3D is of course not the same as that near a circular particlein 2D. Then, the effect of post shape on the transit modesof rigid spherical particles is examined including diamond,square, and triangular posts. The simulation results confirmthat a deterministic picture of the three modes describedabove also holds for the DLD devices with non-circularpost shapes. However, the transition from displacement tozigzag mode may shift for different post geometries incomparison with the circular ones. To describe this transition,an empirical formula (Eq. (8)) by Davis39 is generalizedin order to accommodate the results for other post shapes.Subsequently, similar sets of simulations are performedfor RBCs to investigate the effects of deformability andorientation in relation to the critical diameters for hard spheres.Simulations show that the transit of RBCs through a DLDdevice strongly depends on their deformation and dynamicsin flow, in agreement with previous simulation studies.22–24

Thus, the simple description of particle transit based on thecharacteristic critical size becomes invalid. The predictionsfor particle transit must instead depend on a number ofconditions including device geometry and structure, flow rate,and mechanical properties of the particle. Even though these


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effects significantly complicate such systems, they also leadto new opportunities in particle sorting and detection. Hence,we expect that new sorting designs, which use particle shape,deformability, and other intrinsic properties as a target, can bedeveloped.

II. METHODS AND MODELS

To model fluid flow within a microfluidic device, weemploy the dissipative particle dynamics (DPD) method,25,26

which is a mesoscopic particle-based hydrodynamicsapproach. Suspended particles are represented by a bead-spring model using a ring-like structure. The particles arecoupled to fluid flow by friction forces, which are naturallypart of DPD.

A. Simulation method

DPD25,26 is a mesoscopic simulation technique which hasbeen successfully applied to study various complex fluids.31,32

In DPD, a simulated system consists of a collection of N fluidparticles with mass mi, position ri, and velocity vi and eachindividual particle represents a cluster of atoms or molecules.The dynamics of DPD particles is governed by Newton’ssecond law of motion as follows:

dri = vidt,

dvi =1

mi

j,i

Fi jdt =1

mi

j,i

(FCi j + FDi j + FRi j)dt, (1)

where Fi j is the total force acting between two particles i andj within a selected cutoff radius rc. The total force is a sumof three pairwise forces: conservative (FCi j), dissipative (F

Di j),

and random (FRi j) forces given by

FCi j = ai j(1 − ri j/rc)r̂i j,FDi j = −γi jωD(ri j)(vi j · r̂i j)r̂i j,FRi j = σi jω

R(ri j)ξi jdt−1/2r̂i j,(2)

where vi j = vi − v j, ri j = ri − r j, ri j = |ri j |, and r̂i j = ri j/|ri j |.The coefficients ai j, γi j, and σi j characterize the strength ofthe conservative, dissipative, and random forces, respectively.Both ωD(ri j) and ωR(ri j) are distance-dependent weightfunctions. ξi j is a random number generated from a Gaussiandistribution with zero mean and unit variance.

The dissipative and random forces act together as athermostat to maintain an equilibrium temperature T andgenerate a correct equilibrium Gibbs-Boltzmann distribution.Therefore, they must satisfy the fluctuation-dissipationtheorem26 given by the conditions ωD(ri j) = [ωR(ri j)]2 andσ2 = 2γkBT . In the original DPD method, the weight functionhas been chosen as ωR(ri j) = (1 − ri j/rc)k, with k = 1, whiledifferent choices for this exponent have been made in otherstudies33,34 in order to increase the viscosity of the DPDfluid. The viscosity of a DPD fluid has been calculatedusing a reverse-Poiseuille flow setup,35 where the flow intwo halves of a periodic computational domain is driven inopposite directions. Equations of motion (1) are integratedusing the velocity-Verlet algorithm.36 Table I presents theDPD parameters used in our simulations.

TABLE I. DPD fluid parameters used in simulations. Mass and length forDPD fluid are measured in units of the fluid particle mass m and the cutoffradius rc. n is the fluid’s number density, a is the repulsive-force coefficient,γ is the dissipative-force coefficient, k is the weight-function exponent, andη is the fluid’s dynamic viscosity. In all simulations, we have set m = 1,rc = 1.5, and the thermal energy kBT = 1.

nr2c arc/kBT γrc/√mkBT k ηr

2c/√mkBT

11.25 60 30 0.15 325

B. Suspended particles

We employ 2D simulations, where both rigid circularparticles and RBCs are modeled as closed bead-spring chainswith Nv particles connected by Ns = Nv springs. The springpotential is given by

Vsp =Nsj=1

kBTlm4p

(3x2j − 2x3j)(1 − x j) +

kpl j

, (3)

where l j is the length of the spring j, lm is the maximum springextension, x j = l j/lm ∈ (0,1), p is the persistence length, kBTis the energy unit, and kp is the spring constant. A balancebetween the two force terms in Eq. (3) leads to a nonzeroequilibrium spring length l0. Alternatively, a selected l0 definesa ratio between the coefficients p and kp, while their absolutevalues determine the spring strength such that the Young’smodulus of this spring is given by

Y = l0

(∂2Vsp∂l2

)|l=l0 =

kBT x0p

(1

2(1 − x0)3 + 1)+

2kpl20

, (4)

where x0 = l0/lm.To incorporate membrane bending rigidity, a bending

energy29 is applied as

Vbend =Nsj=1

kb[1 − cos(θ j)], (5)

where kb is the bending constant and θ j is the instantaneousangle between two adjacent springs sharing the commonvertex j. Furthermore, an area constraint is imposed for eachring polymer given by

Varea =ka2(A − A0)2, (6)

where ka is the area constraint coefficient, A0 is the desiredenclosed area, and A is the instantaneous area.

The combination of the targeted area A0, the contourlength L0 = Nsl0, and the bending constant kb controls theshape and rigidity of cells and circular particles. A circularparticle is characterized by a diameter DSP = L0/π. To make acircular particle nearly rigid, its targeted area A0 has been set to4% larger than the area of a circle with the contour length L0.This choice results in a considerable membrane tension due tothe competition of spring and area-constraint forces makingthe particle virtually non-deformable. A RBC is characterizedby the effective diameter DRBC = L0/π and the reducedarea A∗ = 4A0/(πD2RBC) = 0.46 leading to a typical biconcaveshape. The other particle parameters used in simulations aregiven in Table II.


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TABLE II. Model parameters for suspended particles. Nv is the number of vertices, lm is the maximum springextension, l0 is the equilibrium spring length, and D is a characteristic particle diameter which denotes DSP orDRBC. κ is the macroscopic bending rigidity and κ = kbl0 with kb being the bending constant, Y is the Young’smodulus, A0 is the targeted area, and ka is the area constraint coefficient.

Nv lm/l0 κ/(kBTl0) YD/kBT A0/D2 kaD2/kBTCircles 30 . . .60 2.2 500 180 000 1.04π/4 97 000RBCs 50 2.2 50 9 000 0.36 37 430

C. Coupling between fluid and suspended particles

Coupling between the fluid flow and suspended particlesis achieved through viscous friction using the dissipative (FD)and random (FR) DPD forces. The repulsive-force coefficientfor the coupling interactions is set to zero. The strength γ ofthe dissipative force is computed such that no-slip BCs areenforced. The derivation of γ is based on the idealized caseof linear shear flow over a part of a membrane with the lengthL. In a continuum description, the total shear force exerted bythe fluid on the length L is equal to Lηγ̇, where γ̇ is the localwall shear-rate. The same fluid force has to be also transmittedonto a discrete surface structure having NL vertices within thelength L. The force on a single vertex exerted by the fluid canbe found as FA =

Ah

ng(r)FDdA, where n is the fluid numberdensity, g(r) is the radial distribution function of fluid particleswith respect to the membrane particles, and Ah is the half circlearea of fluid above the membrane. The total shear force onthe length L is equal to NLFA. The equality NLFA = Lηγ̇results in an expression of the dissipative force coefficient interms of the fluid density and viscosity, wall density NL/L,and rc. Under the assumption of linear shear flow, the shearrate γ̇ cancels out. This formulation results in satisfaction ofthe no-slip BCs for the linear shear flow over a flat surface;however, it also serves as an excellent approximation forno-slip at the membrane surface.37 Since the conservativeinteractions are turned off, the radial distribution function isstructureless such that g(r) = 1.

D. Simulation setup

A DLD device is simulated using a single obstacle andsuspended particle within the computational domain. This ispossible due to periodic repetition of the device geometry asillustrated in Fig. 1(b). Thus, we employ periodic boundaryconditions (BCs) both in the flow and row-shift directionsdenoted by x and y, respectively. However, for the BCs in x, ashift in the y direction is necessary in order to mimic the shiftbetween two consecutive rows of posts. Here, such a shift isintroduced for every boundary-crossing interaction or event.

Several obstacle geometries are considered, includinga circle, a square, a diamond, and a triangle depicted inFig. 1(a). Figure 1(a) also defines the center-of-mass forthese post geometries and their sizes denoted by W andH , W = H = 15 µm in our simulations. No-slip wall BCsare modeled by a layer of frozen particles with a thicknessof rc whose equilibrium structure characterized by the radialdistribution function is the same as that of the suspending fluid.This minimizes problems with near-wall particle interactions(e.g., significant fluid-density fluctuations near the wall) which

may occur due to an improper distribution of conservativeforces. To prevent wall penetration, fluid particles as wellas suspended structures are subject to reflection at the fluid-solid interface. Bounce-back reflections are employed, sincethey provide a better approximation for the no-slip BCs incomparison to specular reflection of particles. To ensure thatno-slip BCs are strictly satisfied, we also add a tangentialadaptive shear force38 which acts on the fluid particles in anear-wall layer of thickness rc.

The flow in the x direction is driven by a force appliedto each fluid particle. The force value can be tuned to obtaina flow rate of interest. The shift between consecutive rowsresults in a nonzero net flow in the y direction. However,real DLD devices have a limited width and the net flow inthe y direction is prohibited by its side walls. To mimic thissituation, we have also introduced a force in y direction whichcan adapt to follow the condition of no net-flow in the ydirection. Thus, the force in x direction controls the flowrate, while the force in y direction ensures no net flow in ydirection.

In order to relate simulation and physical parameters, weneed to define a time scale. For the case of rigid particles, atime scale can be defined through the fluid viscosity. However,in this case under the assumption of Stokes flow, different flowrates lead to flow profiles which can be scaled to each otherdue to the linearity of the Stokes equation. Thus, the timescaling here is not crucial. In case of flowing RBCs, differentflow rates would result in different cell deformations and needto be looked at separately. In order to define a time scale inthis case, we use a characteristic RBC relaxation time definedas

τ =ηD3RBC

κ. (7)

Using this time scale we can, for instance, relate simulationand physical flow rates. Typical values for healthy RBCs areDRBC = 6.1 µm, η = 1.2 × 10−3 Pa s is the viscosity of bloodplasma, and κ lies within the range of 50–70kBT for thephysiological temperature T = 37 ◦C.

III. RESULTS AND DISCUSSION

A. Separation of rigid spherical particles in circularpost arrays

The dynamics of rigid spherical particles flowing inconventional circular-post arrays is rather well understood.Two transport modes, zigzag and displacement, wereexperimentally observed depending on particle size.7 Toestimate the critical particle size Dc for the transition between


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the two transport modes, an empirical formula has beensuggested based on results of a systematic experimentalstudy39 by varying the gap size and particle diameter andassuming a parabolic flow profile between two neighboringposts. The proposed formula is given by39

Dc = 1.4Gε0.48, (8)

where ε is the row-shift fraction defined as ε = ∆λ/λ. Inthe current study, 2D mesoscopic simulations are employedto investigate the effect of post shapes on the separation ofspherical and non-spherical particles. Before the explorationof a more complex geometry and structure of posts andparticles, careful testing and validation of the simulatedsystem are essential. For this purpose, we have studiedthe separation of rigid spherical particles with 7 differentdiameters (DSP ∈ [2.78,5.59] µm) in circular post arraysby varying the row shift. A detailed comparison betweenour simulation results and the predictions obtained from theempirical formula, Eq. (8), is performed in order to assess thequality of simulation predictions.

Figure 2(a) presents typical trajectories of a sphericalparticle with DSP = 5.12 µm in circular post arrays at differentrow shifts. It is clear that the particle moves according to the

FIG. 2. (a) Typical trajectories of a 2D rigid spherical particle with DSP= 5.12 µm in the conventional circular post arrays at different row shifts∆λ. (b) The separation index Is, which is defined as the ratio of the lateraldisplacement of particles per post to the row shift, for the correspondingparticle sorting as a function of the row-shift fraction. Error bars are indicatedfor all data points.

displacement mode at low row shifts (∆λ . 2.5 µm) andfollows the zigzag mode for larger row shifts (∆λ & 3.0 µm).However, it is interesting to note that at some intermediatevalues of the row shift, particle trajectories exhibit an irregularalternation between displacement and zigzag motions, whichwill be referred to as “mixed mode” subsequently. Similarmotion has been also observed in experiments2,9,10,40 andprevious simulations.9,10 Huang et al.2 suggested that thisbehavior can be attributed to Brownian motion (or diffusion)of particles between different streamlines. We have performedseveral simulations using a larger ambient temperature in DPD(approximately by a factor of two), which leads to an increaseddiffusivity of suspended particles. These results qualitativelyindicate that the mixed-mode region becomes slightly widerin the row-shift fraction. This is not very surprising, since asthe displacement-to-zigzag transition is approached, particlediffusion might make a difference, leading to a widening ofthe transition region (or the mixed-mode region). In contrast,Kulrattanarak et al.9,10 have proposed that the occurrence ofmixed mode is due to the asymmetric flowline distributionbetween posts in certain DLD devices with Gx/Gy ≤ 3 andDpost/Gy > 0.4, where Gx and Gy are the post gap sizesin x and y directions, respectively. The geometry of oursimulated DLD array is in this range with Gx = Gy = G andDpost/Gy = 1.5. It is important to note that this propositionis based on fluid-flow simulations with a point-like tracerparticle; however, for a finite-size particle (e.g., with adiameter comparable to the gap size), the disturbance fieldaround the particle has to be taken into account and maysignificantly contribute to the particle motion within a DLDarray. Furthermore, our simulation method naturally includesthermal fluctuations, and therefore, particle diffusion cannotbe excluded as a potential contribution to the mixed mode.

To quantitatively characterize the motion of particleswithin DLD, we introduce a dimensionless parameter called“separation index” Is, defined as the ratio of the lateraldisplacement of particles per post to the row shift. For theideal displacement mode, Is should be close to unity, sinceparticles are forced to displace laterally along the shift gradientof the post array. For the ideal zigzag mode, Is should be closeto zero as the particles move with the flow without a significantnet lateral displacement. The concept behind this parameter issimilar to the “migration angle”2,9,10 or “separation index”41

used in experiments. Due to the existence of mixed mode,the value of Is in our simulated trajectories is not just 1 or0. As illustrated in Fig. 2(b), there exist a small plateau withintermediate values of Is and some variation, indicating themixed mode. Moreover, the value of Is is found to remainin the range of 0.3–0.6 regardless of the initial position ofa particle and simulation time, which further confirms theexistence of the mixed mode. Away from the mixed region, Isquickly attains values close to 1 or 0. In view of these results,the magnitude of Is ∈ [0.3,0.6] is used as a quantitativemeasure for the mixed mode and the transition to/from zigzagor displacement dynamics. Therefore, Is > 0.6 and Is < 0.3are chosen to define the displacement and zigzag modes,respectively.

The identical strategy and analysis have been employedto assess the separation of all the seven particle diameters in


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circular post arrays. The observed modes as a function of therow shift fraction (ε) and the normalized particle diameter(DSP/G) are summarized in Fig. 3. It is apparent that thecritical particle sizes calculated from the empirical formulaof Eq. (8) approximately reside in the region of mixed modewhich defines the transition from the zigzag to displacementmode. The good agreement between our simulations andexperimental results of Davis39 validates the 2D modeland shows that it is able to capture the essential physicalfeatures required to correctly describe the particle separationin conventional DLD devices with circular posts. Therefore,further simulations with modifications of the post shapesand particle properties should provide reliable and valuableguidelines for the performance of different DLD devices.

B. Effect of post shapes on the separation of rigidspherical particles

Diamond, square, and triangular posts have also beenstudied to explore the effect of post shape on the flow behaviorof rigid spherical particles within DLDs. In comparison withthe circular posts, the common geometric feature of these threeposts is the sharp edges of the post surfaces whose curvature,position, and symmetry all strongly affect the flow propertiesin post gaps. Indeed, significant changes are observed in theflow profiles for different post arrays, as depicted in Fig. 4. Theplots show that the region of the largest flow velocity withinthe post gaps resembles elliptical, circular, cigar-like, andpear-like patterns for circular, diamond, square, and triangularposts, respectively. Moreover, it is worth mentioning that inthe square-post array, there is a small region with nearly zeroflow rate in the gap between adjacent columns. All thesechanges must alter the motion of particles through the postarrays and consequently influence the separation efficiency inDLD devices.

A collection of the different transit modes for rigidspherical particles in the diamond- and square-post arrays isshown in Fig. 5. In comparison with the behavior in circular-

FIG. 3. Mode diagram for rigid spherical particles in circular post arrays asa function of the row shift fraction (ε) and the normalized particle diameter(DSP/G). The predicted critical size from the empirical formula in Eq. (8) isdrawn by the solid line.

FIG. 4. Comparison of flow velocities (x component) in DLD arrays with(a) circular, (b) diamond, (c) square, and (d) triangular posts at the row shiftfraction of ε = 1/8. All velocities in the flow direction (x axis) are normalizedby the maximum value of velocity observed in the square post array. Redregions represent the high flow rate, while the dark blue regions correspondto the zero flow velocity at the boundaries with no-slip condition.

post arrays, the transition from the zigzag to displacementmode is shifted slightly or significantly to a higher valueof the row-shift fraction ϵ in the diamond- or square-postarrays, respectively, indicating a reduction of the critical sizein these devices. These simulation results are consistent withthe experimental and numerical studies.3,4,20,42 The reducedcritical size can be understood intuitively on basis of thechange of the flow profiles induced by the post shapes.According to the theory in Refs. 2, 7, and 39, fluid flow in thepost gap can be split into several (N = 1/ε) streams carryingequal volumetric flow rate. The width D f ,1 of the first streamdirectly next to the post (see Fig. 4) is assumed to correspondto the critical particle radius for the transition from the zigzagto displacement mode. In general, D f ,1 is wider in comparisonto middle streams in order to accommodate more fluid flow,since the first stream is at the wall with no-slip boundaryconditions. For diamond-shaped posts, the existence of sharpedges in comparison to the smooth bending of a circular-postsurface leads to a smaller D f ,1 and a smaller critical particle


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FIG. 5. Mode diagram for the rigid spherical particles in (a) diamond and (b)square post arrays as a function of the row shift fraction (ε) and the normal-ized particle diameter (DSP/G). Critical size from the empirical formula inEq. (8) is drawn by the solid line.

size. In contrast, the flat edges of the square-shaped postshould smoothen the velocity distribution and consequentlyinduce a larger critical size, which is in contrast to ournumerical results and observations in Fig. 5. In light of thisseeming contradiction, we define another stream wherein thefluid moves zigzag-like from the top to the lower channel anddenote the corresponding width as D f ,z (see Fig. 4(c)). Forthe circular, diamond, and triangular posts, D f ,z ≃ D f ,1 hasbeen measured, while for the square posts, D f ,z is found to be10%–20% smaller than D f ,1 independently of the row shift.Hence, a possible reason for a strong reduction of the criticalparticle size in square-post arrays is that the sharp corners ofsquare posts reside in both the inlet and outlet of the post gapand induce a small region with nearly zero flow rate in the gapbetween adjacent post columns. It is clear from Fig. 4 that forthe circular, diamond, and triangular posts, the flow rate in thegap between adjacent post columns is significantly larger thanthat for the square posts. In this way, the original first streamis seriously obstructed, and therefore, a significant reductionof critical particle size is observed for square-post arrays.

Recently, Wei et al.42 proposed a general formula toestimate the critical particle sizes of DLD arrays with different

post shapes by introducing a shape-dependent prefactor toEq. (8). However, this parameter is simply obtained by datafitting as a function of the row shift and a geometric parameterof the gap between adjacent post columns, which is notsufficient to describe the geometric features of post shapes.For example, their numerical studies have shown that theshape factor and the resulting formula appear to be the samefor square- and I-shape posts, which is in contrast with theexperimental observations.20 Additionally, the critical particlesize from their formula42 used for square post arrays is foundto significantly deviate from our predictions in most cases.In view of these differences, we propose a more generalformula

Dc = αGεβ, (9)

where the dimensionless parameters α and β are geometricfactors determined by the shape and arrangement of posts.For the conventional circular posts, α = 1.4 and β = 0.48 asin Eq. (8). For the diamond and square posts in Fig. 5, thebest fitting curves are obtained for the parameters α = 1.4, β= 0.52 and α = 1.4, β = 0.69, respectively. Interestingly, thecoefficient α remains essentially unaffected for circular,diamond, and square posts.

Unlike the symmetric shapes discussed above, thetriangular posts break the top-bottom symmetry, see Fig.1(a). Accordingly, an asymmetric flow profile with a skewedparabolic shape is induced through the gap, as can be seen fromthe pear-like pattern shown in Fig. 4(d). This feature resultsin two distinct critical particle sizes depending on the flowdirection or row shift, which is not the case for the symmetricposts. Alternatively, the change in the flow direction or rowshift for triangular posts is equivalent to turning the triangularposts upside down. As fluid moves forward in the triangular-post array with a positive row shift (see Fig. 1(b)), onecritical particle size should be determined by the stream widthadjacent to the top vertex of a triangle. However, for a reversedflow direction (from right to left) with negative row shift,another critical size should be governed by the stream widthjust above the flat bottom edge, and the corresponding value ofthe critical size can be expected to be smaller than that for theprevious flow direction (from left to right). This tendency isin accordance with our simulation results as illustrated in Fig.6(a). For positive row shift, the transition between the zigzagand displacement modes is shifted to higher values of rowshifts, indicating a reduced critical particle size compared tothat for circular posts. In contrast, an increased critical particlesize is observed for negative row shifts. The fits in Fig. 6(a)using Eq. (9) correspond to parameters α = 1.4, β = 0.44 andα = 1.4, β = 0.61 for the negative and positive row shifts,respectively.

Figure 6(b) presents the separation index Is for triangular-post arrays. The value of Is not only characterizes the modeof particle motion but also provides the physical lateraldisplacement (i.e., Is∆λ) of a particle per single row of posts.The value of Is also provides guidance on the design of a DLDdevice. For instance, in case of the triangular-post array with| ∆λ |= 4 µm (ε = 0.16) in Fig. 6(b), a smaller particle withDSP = 3.72 µm is in the zigzag mode with Is ≈ 0, a particlewith DSP = 4.65 µm is in the mixed mode with Is ≈ 0.5, and


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FIG. 6. (a) Mode diagram for the rigid spherical particles in triangular-postarrays as a function of the row shift fraction (ε) and the normalized particlediameter (DSP/G). (b) Typical behavior of the separation index Is as afunction of the absolute value of the row-shift fraction.

a particle with DSP = 5.12 µm is in the displacement modewith Is ≈ 1. Thus, the choice of such shift with ∆λ = 4 µmwithin a DLD is advantageous for the separation of these threeparticle sizes. An analogous choice for optimal row shifts canbe also made for the case of negative row displacements.An interesting observation in Fig. 6(b) is that the transitionbetween the displacement and zigzag modes appears to bemuch sharper for negative in comparison to positive rowshifts.

C. Motion of RBCs in different DLD arrays

So far we have focused on the motion of rigid sphericalparticles within DLD arrays. However, many bioparticles andcells of interest are non-spherical and deformable, and theirshape and deformability are well known to largely influencetheir trajectories in DLD devices.5,6,19,43 To better understandthe behavior of flexible particles in flow within DLDs, we haveperformed a number of simulations using a 2D RBC modelfor various post shapes including circular-, diamond-, andsquare-post arrays. Since RBCs are deformable, an effectivesize denoted as Deff is defined to characterize the transition ofRBCs from the zigzag to displacement mode. Figure 7 showsthe separation index for RBCs within circular, diamond, and

FIG. 7. Typical distributions of the separation index Is for RBCs withincircular, diamond, and square post arrays as a function of the row shiftfraction.

square post arrays. We find that the transition from the zigzagto displacement mode for RBCs is at ∆λ = 1.51 ± 0.05 µm,1.07 ± 0.18 µm, and 3.46 ± 0.33 µm for circular-, diamond-,and square-post arrays, respectively. According to our fits,these row-shift values correspond to Deff = 3.65 ± 0.07 µm,2.65 ± 0.14 µm, and 3.50 ± 0.15 µm, respectively. Theseeffective sizes lie between the thickness and diameter ofa RBC, indicating that RBCs deform and display differentdynamics in various flow fields induced by the post shapes.

Figures 8 and 9 illustrate how RBCs move and deformin the zigzag and displacement modes in different postarrays. In the zigzag mode (Fig. 8), there is usually oneflipping motion in each zigzag period, which increases theeffective size of RBCs and in turn facilitates the jump ofRBCs from the top to the bottom stream next to the post.This process can be occasionally accomplished by a dramaticdeformation, as depicted in the first zigzag motion in Fig. 8(c).After that, RBCs usually slip smoothly past the subsequentposts by slightly deforming along the geometry of a post. IfDeff > 2D f ,1, RBCs enter the adjacent stream and flow abovethe next post, resulting in the displacement motion depictedin Fig. 9. It is important to note that when Deff ≫ 2D f ,1 in thesquare-post array, the displacement motion of RBCs changesfrom slipping around the posts (see Figs. 8(a) and 8(b)) toavoiding any contact with the post and moving quickly withan orientation parallel to the gradient of array (Fig. 8(c)). Thisnovel behavior is due to the cigar-like flow profile in the gap,which resembles flow in a short cylindrical microchannel. Inmicrochannel flow, RBCs migrate away from the channel wallsdue to hydrodynamic interactions of RBCs with the walls,which is often referred to as lift force.44–46 If Deff < 2D f ,1,RBCs move along the zigzag path of the first stream. Here,the flow profile in the gap between the adjacent post columnsplays an important role in deforming RBCs and consequentlyinfluences their flow path. As an example, for the case ofsquare posts in Fig. 8(c), after slipping past the post, RBCssubside gradually from the top channel and flip into thelower channel, which is clearly different from the motions incircular- and diamond-post arrays.


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FIG. 8. Snapshots of typical zigzag tra-jectories of RBCs in DLD arrays with(a) circular, (b) diamond, and (c) squareposts at ∆λ = 5.0 µm. The regular alter-nation of black and red RBCs representssequential frames with an interval of0.023τ.

To describe the shape and morphology change of RBCsin DLD arrays, a parameter called asphericity δ is used tomeasure the deviation from spherical (or circular) geometry,

δ =(λ1 − λ2)2(λ1 + λ2)2 , (10)

where λ1 and λ2 are the square root of two non-zeroeigenvalues of the squared radius-of-gyration tensor. Thevalue of δ varies from 0 to 1, corresponding to a perfectcircle and a strongly elongated object, respectively. For abiconcave 2D RBC shape in equilibrium, the asphericity isequal to δ ≈ 0.29. Following this definition, Fig. 10 presentsthe morphology change of RBCs under different conditionsin terms of a probability density. It is apparent that withthe transition from the displacement to zigzag mode with

increasing row shift, RBCs are subject to more deformationin both circular- and square-post arrays, as indicated by thehigher probability for lower asphericities shown in Figs. 10(a)and 10(c). However, RBCs display significant deformation inboth zigzag and displacement modes, which is also confirmedby the flipping motion in Figs. 8(b) and 9(b).

In view of these observations, a zigzag period can bedivided into two stages: (i) particle motion across the gaptoward the first stream above the post and (ii) flowing alongthe zigzag path of the stream. For rigid spherical particles,it is a straightforward and deterministic process based onparticle size. For soft particles or cells however, it canbe very complicated, since the deformation and motion ofparticles are highly sensitive to the flow field within thepost array. This also indicates that the flow rate in a DLD

FIG. 9. Snapshots of typical displace-ment trajectories of RBCs in DLDarrays with (a) circular posts at ∆λ= 1.25 µm, (b) diamond posts at ∆λ= 1 µm, and (c) square posts at ∆λ= 1.25 µm. The regular alternation ofblack and red RBCs corresponds tosequential frames with an interval of0.023τ.


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FIG. 10. Distributions of the asphericity (δ) of RBCs in DLD arrays with (a)circular, (b) diamond, and (c) square posts for various row shifts.

device becomes an additional parameter to consider for theseparation of soft particles and cells. The effect of flow rate onthe RBC motion within a DLD device with circular posts hasbeen already investigated experimentally,19 showing a strongdependence. In another experimental study, it has been foundthat a rotational motion of a non-isotropic particle induced byI-shape posts might be advantageous for particle sorting.20 Asimilar rotational motion has been also observed in our studyfor RBCs in particular for diamond- and square-shaped posts.As a conclusion, the notion of a single critical size for particleseparation, as suggested for the rigid spherical particles, islikely to be an oversimplified concept when applied to non-isotropic, deformable particles and cells. Deformable particlesand cells clearly show very rich dynamics in DLD devices,whose underling mechanisms still need to be further exploredin order to reach the understanding required for a rationaldesign of DLD devices for sorting.

IV. SUMMARY AND CONCLUSIONS

We have presented a systematic study of particle motionwithin DLD arrays using a coarse-grained mesoscopic

modeling approach in two spatial dimensions. Several obstaclegeometries including circular, diamond, square, and triangularposts and particle sizes were considered. The simulationresults for rigid spherical particles in circular-post arrays arefound to be in a very good agreement with the availableexperimental data. The dynamics of spherical particles canbe divided into the three major modes: displacement, mixed,and zigzag. The displacement mode is characterized by themotion where a particle simply follows the gradient of rowshift within a device with a separation index close to unity. Thezigzag mode corresponds to the particle motion with nearlyzero lateral displacement within the device, which can beidentified by a nearly zero separation index. The mixed moderesults in an intermediate state with alternating displacementand zigzag sections and can be characterized by the separationindex Is ∈ [0.3,0.6]. The transition from the displacement tozigzag mode in circular-post arrays is well captured by anempirical formula (Eq. (8)), which defines a critical particlesize.

Simulations of diamond-, square-, and triangular-postarrays display significant differences in particle critical sizein comparison with the circular-post devices. The flow fieldstrongly depends on the geometry of a device and consequentlymodifies particle trajectories in the corresponding post arrays.However, the behavior of a rigid spherical particle in thesedevices can still be well described by the three characteristicmodes, and the transition from the displacement to the zigzagmode is well captured by a generalized empirical formula(Eq. (9)) with two fitting parameters. The possibility of aunified empirical description for the transition indicates thatqualitatively, the same flow mechanisms are of importancein such devices, and all of them can be potentially usedfor the sorting of rigid spherical particles with differentsizes. However, the flow details also matter when fine tuningbecomes important and some geometries might be preferabledepending on the problem of interest.

Finally, the flow behavior of deformable particles suchas RBCs has been investigated for different post geometries.The simulations have shown that RBCs experience strongdeformations due to fluid flow, which determine theircorresponding mode of motion. In comparison to rigidspherical particles, the transition from the displacement tothe zigzag mode becomes very broad with a significant partdevoted to the mixed mode. Also, the dynamics of RBCs (e.g.,tumbling) in flow plays a significant role in determining theirtraversal trajectory through a DLD device. Both dynamics anddeformation strongly depend on the device geometry, since itis governed by local flow fields. In addition, it is expected thatdifferent flow rates within a device should lead to distinctbehaviors of RBCs, making this an important parameterfor consideration. These results indicate that a simplifieddescription with a single critical size becomes invalid in caseof non-isotropic and deformable particles and/or cells. Thebehavior of deformable particles in DLD devices is muchricher in comparison with the rigid spherical particles, whichnot only brings additional complications but also providesnew opportunities. Thus, it is plausible to expect that non-isotropic and deformable particles can be sorted based ontheir deformability, geometry, and potentially other intrinsic


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properties. We expect that this research direction will receiveconsiderable attention in the near future.

ACKNOWLEDGMENTS

We would like to thank Stefan H. Holm, Jason P. Beech,and Jonas O. Tegenfeldt (Lund University, Sweden) forstimulating discussions. We acknowledge the FP7-PEOPLE-2013-ITN LAPASO “Label-free particle sorting” for financialsupport. Dmitry A. Fedosov acknowledges funding by theAlexander von Humboldt Foundation. We also gratefullyacknowledge a CPU time grant by the Jülich SupercomputingCenter.

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behavior of rigid and deformable particles in deterministic lateral … · 2018. 6. 19. · preface...

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